Final answer:
To solve the given initial value problem y'-y=10te2t, we can use the method of integrating factor. The solution to the initial value problem is y = (5t^2 - 2t + Ce^t)e^t, where C is a constant determined by the initial condition.
Step-by-step explanation:
To solve the given initial value problem y'-y=10te2t, we can use the method of integrating factor. First, we rewrite the equation in the form y' - y = 10te^(2t). The integrating factor is e^(-t). Multiplying both sides of the equation by the integrating factor gives us e^(-t)y' - e^(-t)y = 10te^(t). We now have (e^(-t)y)' = 10te^(t), which can be integrated to find y. Integrate both sides to get e^(-t)y = ∫(10te^t)dt. Solving the integral yields e^(-t)y = (5t^2 - 2t)e^t + C, where C is a constant of integration. Finally, divide both sides by e^(-t) to solve for y. The solution to the initial value problem is y = (5t^2 - 2t + Ce^t)e^t, where C is a constant determined by the initial condition.